All AdS_7 solutions of type II supergravity

TL;DR

This work broadens the landscape of AdS7 solutions by classifying AdS7 × M3 backgrounds in type II supergravity, showing IIB imposes no solutions while massive IIA with Romans mass F0 admits many. Using a pure spinor formalism akin to generalized complex geometry, the authors derive a complete set of SUSY equations that fix the internal geometry to an S^2 fibration over an interval, with the warping and fluxes determined by a pair of coupled ODEs. The global structure requires M3 to be S^3, achievable via appropriate boundary conditions and, in some cases, D8-brane stacks that regularize the geometry. Explicit numerical solutions illustrate regular configurations with D8-branes and discuss their relation to six-dimensional (1,0) theories, highlighting a rich class of holographic duals and brane setups. Overall, the paper provides a systematic, Ansatz-free construction and classification of AdS7 × M3 solutions in IIA that enriches the holographic dictionary for 6d SCFTs.

Abstract

In M-theory, the only AdS_7 supersymmetric solutions are AdS_7 x S^4 and its orbifolds. In this paper, we find and classify new supersymmetric solutions of the type AdS_7 x M_3 in type II supergravity. While in IIB none exist, in IIA with Romans mass (which does not lift to M-theory) there are many new ones. We use a pure spinor approach reminiscent of generalized complex geometry. Without the need for any Ansatz, the system determines uniquely the form of the metric and fluxes, up to solving a system of ODEs. Namely, the metric on M_3 is that of an S^2 fibered over an interval; this is consistent with the Sp(1) R-symmetry of the holographically dual (1,0) theory. By including D8 brane sources, one can numerically obtain regular solutions, where topologically M_3 = S^3.

Figures (5)

• Figure 1: A plot of the vector field induced by (\ref{['eq:odeA']}) on $\{\tilde{\phi} \equiv \phi + A, x\}$, for $F_0=40/2\pi$ (in agreement with flux quantization, (\ref{['eq:F0n0']}) below). The green circle represents the point $\{\phi+A=\log(4/F_0),x=1\}$, whose role will become apparent in section \ref{['sub:loc']}. The dashed line represents the locus along which the denominators in (\ref{['eq:odeA']}) vanish.
• Figure 2: Massless solution in IIA. We show here the radius of the $S^2$ (orange), the warping factor $e^{2A}$ (black; multiplied by a factor $1/20$), and the string coupling $e^\phi$ (green; multiplied by a factor $5$). We see that the warping goes to zero at the two poles. The angular coefficient of the orange line can be seen to be $3/4$ as in (\ref{['eq:nearD6']}). The two singularities are due to $k$ D6 and $k$ anti-D6 (in this picture, $k=20$).
• Figure 3: Solution for $F_0=40/2\pi$. We imposed regularity at the north pole, and evolved towards positive $r$. In \ref{['fig:nod8-a']} we again plot the radius of the $S^2$ (orange), the warping factor $e^{2A}$ (black; multiplied by a factor $1/20$), and the string coupling $e^\phi$ (green; multiplied by a factor $5$). With increasing $r$, the plot gets more and more similar to the one for the massless case in figure \ref{['fig:massless']}. There is a stack of D6's at the south pole (in this picture, $k=112$ of them), as in the massless case, although this time it also has a diverging NS three-form $H$. Notice that the size of the $S^2$ goes linearly near both poles, but with angular coefficients $1$ near the north pole (appropriate for a regular point) and $3/4$ for the south pole (appropriate for a D6, as seen in (\ref{['eq:nearD6']})). In \ref{['fig:nod8-b']}, we see the path described by the solution in the $\{ A+ \phi, x\}$ plane, overlaid to the vector field shown in figure \ref{['fig:stream']}.
• Figure 4: Regular solution with one D8 stack. Its position can be seen in the graph as the value of $r$ where the derivatives of the functions jump; it is fixed by (\ref{['eq:jump']}). In \ref{['fig:1d8-a']} we again plot the radius of the $S^2$ (orange), the warping factor $e^{2A}$ (black; rescaled by a factor $1/20$), and the string coupling $e^\phi$. We also plot $\frac{1}{\pi}\hat{b}(r)=\frac{1}{4\pi^2}\int_{S^2_r} B_2$ (dashed, light green); to guide the eye, we have periodically identified it as described in section \ref{['sub:loc']}. (The apparent discontinuities are an artifact of the identification.) The fact that it starts and ends at $\hat{b}=0$ is in compliance with flux quantization for $H$; we have $\frac{1}{4\pi^2}\int H= -5$. The flux parameters are $\{n_0,n_2\}=\{10,-50\}$ on the left (namely, near the north pole), $\{-40, 0\}$ on the right (near the south pole). In \ref{['fig:2d8-b']}, we see the path described by the solution in the $\{ A+ \phi, x\}$ plane, overlaid to the relevant vector field, that this time changes with $n_0$.
• Figure 5: Regular solution with two D8 stacks. As in figure \ref{['fig:1d8']}, their positions are the two values of $r$ where the derivatives of the functions jump. In \ref{['fig:2d8-a']} we again plot the radius of the $S^2$ (orange), the warping factor $e^{2A}$ (black; rescaled by a factor $1/20$), and the string coupling $e^\phi$ (green; rescaled by a factor $5$), and $\hat{b}$ (as in figure \ref{['fig:1d8']}; this time $\frac{1}{4\pi^2}\int H= -3$). The flux parameters are: $\{n_0,n_2\}=\{40,0\}$ on the left (namely, near the north pole); $\{0,-40\}$ in the middle; $\{-40,0\}$ on the right (near the south pole). The region in the middle thus has $F_0=0$; it is indeed very similar to the massless case of figure \ref{['fig:massless']}. In \ref{['fig:2d8-b']}, we see the path described by the solution in the $\{ A+ \phi, x\}$ plane, overlaid to the relevant vector field, that again changes with $n_0$.